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Ch.12 - Solids and Modern Material
Tro - Chemistry: A Molecular Approach 4th Edition
Tro4th EditionChemistry: A Molecular ApproachISBN: 9780134112831Not the one you use?Change textbook
All textbooksTro 4th EditionCh.12 - Solids and Modern MaterialProblem 32
Chapter 12, Problem 32

What is the packing efficiency of a face-centered cubic unit cell? Please show your work.

Verified step by step guidance
1
Understand that packing efficiency is the fraction of volume in a crystal structure that is occupied by the constituent particles (atoms, ions, or molecules).
Recognize that in a face-centered cubic (FCC) unit cell, there are 4 atoms per unit cell. This is calculated as: 8 corner atoms × 1/8 (each corner atom is shared by 8 unit cells) + 6 face atoms × 1/2 (each face atom is shared by 2 unit cells) = 4 atoms.
Recall that the volume of a sphere (atom) is given by \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the atom.
Calculate the volume of the FCC unit cell. The edge length \( a \) of the FCC unit cell is related to the atomic radius \( r \) by the equation \( a = 2\sqrt{2}r \). Therefore, the volume of the unit cell is \( a^3 = (2\sqrt{2}r)^3 \).
Determine the packing efficiency by dividing the total volume of the atoms in the unit cell by the volume of the unit cell, and then multiply by 100 to express it as a percentage. The formula is: \( \text{Packing Efficiency} = \left( \frac{4 \times \frac{4}{3} \pi r^3}{(2\sqrt{2}r)^3} \right) \times 100 \).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Face-Centered Cubic (FCC) Structure

The face-centered cubic (FCC) structure is a type of crystal lattice where atoms are located at each of the corners and the centers of all the cube faces. This arrangement allows for a high packing density, as each unit cell contains four atoms. Understanding the geometry of the FCC unit cell is essential for calculating packing efficiency.
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Face Centered Cubic Example

Atomic Radius and Unit Cell Dimensions

In an FCC unit cell, the relationship between the atomic radius (r) and the edge length (a) is crucial for calculations. The face diagonal of the cube can be expressed in terms of the atomic radius as √2a = 4r. This relationship helps determine the dimensions of the unit cell, which are necessary for calculating the volume occupied by the atoms.

Packing Efficiency Calculation

Packing efficiency is defined as the fraction of volume in a crystal structure that is occupied by atoms. For an FCC unit cell, the packing efficiency can be calculated using the formula: (Volume of atoms in the unit cell) / (Volume of the unit cell). In FCC, this results in a packing efficiency of approximately 74%, indicating that a significant portion of the unit cell is filled with atoms.
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Related Practice
Textbook Question

Molybdenum crystallizes with the body-centered unit cell. The radius of a molybdenum atom is 136 pm. Calculate the edge length of the unit cell and the density of molybdenum

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Textbook Question

Rhodium has a density of 12.41 g/cm3 and crystallizes with the face-centered cubic unit cell. Calculate the radius of a rhodium atom.

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Textbook Question

Determine the number of atoms per unit cell for each metal.

(b) Tungsten

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Textbook Question

Platinum crystallizes with the face-centered cubic unit cell. The radius of a platinum atom is 139 pm. Calculate the edge length of the unit cell and the density of platinum in g/cm3.

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Textbook Question

Determine the number of atoms per unit cell for each metal.

(a) Polonium

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